How do you divide ##(3x ^ { 2} + 7x + 4) div ( x + 1)##?

3x+4
Let’s first define two terms divisor and dividend
divisor = x+1& dividend = ##3x^2 +7x + 4##
Let us begin and this will be difficult explaining w/o being able to write with a pen on paper. I’ll try to go through the steps mathematically and hope that you can follow along visually.
##x+1// 3x^2+7x+4##

Use your highest order variable in the divisor to divide into the first term of the dividend. We want to ask x times what will give us something to cancel out that ##3x^2## when its being subtracted as per long division.

We can write 3x on top of the division bar as our first term and multiply 3x by (x+1) giving us ##3x^2 +3x## but now we must subtract those values underneath the ##3x^2 +7x## portion of the dividend.

This is extremely difficult to write without being able to draw what’s happening

It would look something like this:
##3x^2+7
-3x^2-3x= 4x##
Now we drop down the last term/constant the 4 so that it should read something like 4x+4 under the first subtraction under the dividend. Next we ask what value or variable can we multiply our divisor with to cancel out the 4x? Well if we just use the number 4, it will yield us a perfect ##(4x+4)## and we would write on quotient part of the division: 3x * +4 *.
Ultimately you would just subtract 4x+4 by -(4x+4) which results in zero.
Thus the answer is ##3x +4##
Please note that your teacher may teach you guys this method or , which may be easier but this method always works if you follow the steps. I personally always used the standard way since I would mix up the +/- signs and would get questions wrong.
On the bright side you never have to do these operations in Calculus and above so just learn and appreciate it while in Precalc :).
I hope all this helped and made sense!